In this work we present a formulation of the Teukolsky equation forNumerical solution of the 2+1 Teukolsky equation, application to late-time decays

We quantify the consistency of numerical-relativity black-hole-binary waveforms for use in gravitational-wave (GW) searches with current and planned ground-based detectors. We compare previously published results for the ð' ¼ 2; jmj ¼ 2Þ mode of the gravitational waves from an equalmass nonspinning binary, calculated by five numerical codes. We focus on the 1000M (about six orbits, or 12 GW cycles) before the peak of the GW amplitude and the subsequent ringdown. We find that the phase and amplitude agree within each code's uncertainty estimates. The mismatch between the ð' ¼ 2; jmj ¼ 2Þ modes is better than 10 À3 for binary masses above 60M with respect to the Enhanced LIGO detector noise curve, and for masses above 180M with respect to Advanced LIGO, Virgo, and Advanced Virgo. Between the waveforms with the best agreement, the mismatch is below 2 Â 10 À4 . We find that the waveforms would be indistinguishable in all ground-based detectors (and for the masses we consider) if detected with a signal-to-noise ratio of less than % 14, or less than % 25 in the best cases.

Abstract. It is well-known that Bowen-York initial data contain spurious radiation. Although this "junk" radiation has been seen to be small for non-spinning black-hole binaries in circular orbit, its magnitude increases when the black holes are given spin. It is possible to reduce the spurious radiation by applying the puncture approach to multiple Kerr black holes, as we demonstrate for examples of head-on collisions of equal-mass black-hole binaries.

We investigate the stationary end state obtained by evolving a collapsing spherical star with the gauges routinely adopted to study puncture black holes. We compare the end state of the collapse with the trumpet solution found in the evolution of a single wormhole slice and show that the two solutions closely agree. We demonstrate that the agreement is caused by the use of the Gamma-driver shift condition, which allows the matter to fall inwards into a region of spacetime that is not resolved by the numerical grid, and which simultaneously finds the stationary coordinates of the trumpet outside the matter.

Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally, a derivation of Newtonian Gravity from Einstein's Equations is given.